Higher order expansions for the probabilistic local Cauchy theory of the cubic nonlinear Schrödinger equation on $\mathbb{R}^3$

We consider the cubic nonlinear Schrödinger equation (NLS) on

R 3 \mathbb {R}^3

with randomized initial data. In particular, we study an iterative approach based on a partial power series expansion in terms of the random initial data. By performing a fixed point argument around the second order expansion, we improve the regularity threshold for almost sure local well-posedness from our previous work. We further investigate a limitation of this iterative procedure. Finally, we introduce an alternative iterative approach, based on a modified expansion of arbitrary length, and prove almost sure local well-posedness of the cubic NLS in an almost optimal regularity range with respect to the original iterative approach based on a power series expansion.